A000726 -id:A000726 - cp's OEIS frontend

A257989 The crank of the partition having Heinz number n.

-1, 2, -2, 3, 0, 4, -3, 2, 0, 5, -2, 6, 0, 3, -4, 7, 1, 8, -1, 4, 0, 9, -3, 3, 0, 2, -1, 10, 1, 11, -5, 5, 0, 4, -2, 12, 0, 6, -3, 13, 1, 14, -1, 3, 0, 15, -4, 4, 1, 7, -1, 16, 2, 5, -2, 8, 0, 17, -1, 18, 0, 4, -6, 6, 1, 19, -1, 9, 1, 20, -3, 21, 0, 3, -1, 5, 1, 22, -4, 2, 0, 23, -1, 7, 0, 10, -2, 24, 2, 6, -1

Offset: 2

Emeric Deutsch, May 18 2015

sign

The crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's).
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n, the subprogram b yields the number of 1's in the partition with Heinz number n and the subprogram c yields the number of parts that are larger than the number of 1's in the partition with the Heinz number n.

			a(12) = - 2 because the partition with Heinz number 12 = 2*2*3 is [1,1,2], the number of parts larger than the number of 1's is 0 and the number of 1's is 2; 0 - 2 = -2.
a(945) = 4 because the partition with Heinz number 945 = 3^3 * 5 * 7 is [2,2,2,3,4] which has no part 1; the largest part is 4.
From _Gus Wiseman_, Apr 05 2021: (Start)
The partitions (center) with each Heinz number (left), and the corresponding terms (right):
   2:    (1)    -> -1
   3:    (2)    ->  2
   4:   (1,1)   -> -2
   5:    (3)    ->  3
   6:   (2,1)   ->  0
   7:    (4)    ->  4
   8:  (1,1,1)  -> -3
   9:   (2,2)   ->  2
  10:   (3,1)   ->  0
  11:    (5)    ->  5
  12:  (2,1,1)  -> -2
  13:    (6)    ->  6
  14:   (4,1)   ->  0
  15:   (3,2)   ->  3
  16: (1,1,1,1) -> -4
(End)

Alois P. Heinz, Table of n, a(n) for n = 2..10000
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
B. C. Berndt, H. H. Chan, S. H. Chan, W.-C. Liaw, Cranks and dissections in Ramanujan's lost notebook, J. Comb. Theory, Ser. A, 109, 2005, 91-120.
B. C. Berndt, H. H. Chan, S. H. Chan, W.-C. Liaw, Cranks - really the final problem, Ramanujan J., 23, 2010, 3-15.
G. E. Andrews, K. Ono, Ramanujan's congruences and Dyson's crank, Proc. Natl. Acad. Sci. USA, 102, 2005, 15277.
FindStat, Dyson's crank of a partition.
K. Mahlburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Natl. Acad. Sci. USA, 102, 2005, 15373-15376.
Wikipedia, Crank of a partition

Cf. A215366, A257988.
Indices of zeros are A342192.
A001522 counts partitions of crank 0.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
Cf. A000005, A000726, A056239, A112798, A124010, A224958, A325351, A325352.

with(numtheory): a := proc (n) local B, b, c: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do; [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: b := proc (n) if `mod`(n, 2) = 1 then 0 else 1+b((1/2)*n) end if end proc: c := proc (n) local b, B, ct, i: b := proc (n) if `mod`(n, 2) = 1 then 0 else 1+b((1/2)*n) end if end proc: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for i to bigomega(n) do if b(n) < B(n)[i] then ct := ct+1 else  end if end do: ct end proc: if b(n) = 0 then max(B(n)) else c(n)-b(n) end if end proc: seq(a(n), n = 2 .. 150);

B[n_] := Module[{nn, j, m}, nn =  FactorInteger[n]; For[j = 1, j <= Length[nn], j++, m[j] = nn[[j]]]; Flatten[Table[Table[PrimePi[m[i][[1]]], {q, 1, m[i][[2]]}], {i, 1, Length[nn]}]]];
b[n_] := b[n] = If[OddQ[n], 0, 1 + b[n/2]];
c[n_] := Module[{ct, i}, ct = 0; For[i = 1, i <= PrimeOmega[n], i++, If[ b[n] < B[n][[i]], ct++]]; ct];
a[n_] := If[b[n] == 0, Max[B[n]], c[n] - b[n]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Apr 25 2017, after Emeric Deutsch *)
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Table[ck[primeMS[n]],{n,2,30}] (* Gus Wiseman, Apr 05 2021 *)

A266648 Expansion of Product_{k>=1} (1 + x^(3*k)) / (1 - x^k).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 21, 31, 46, 64, 89, 126, 170, 231, 314, 417, 552, 733, 955, 1244, 1617, 2079, 2665, 3413, 4331, 5485, 6931, 8704, 10901, 13629, 16949, 21033, 26045, 32123, 39529, 48553, 59429, 72599, 88518, 107624, 130599, 158209, 191175, 230611, 277717, 333730, 400375, 479598, 573386, 684481

Offset: 0

Vaclav Kotesovec, Jan 02 2016

nonn

a(n) is the number of overpartitions wherein only parts that are a multiple of three may be overlined. - Alois P. Heinz, Feb 03 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020, p. 6.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Cf. A000726, A015128, A100405, A266647, A266649, A266650, A285445, A285447.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(irem(i, 3)=0, 2, 1)*add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2025

nmax = 50; CoefficientList[Series[Product[(1+x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(7) * exp(sqrt(7*n)*Pi/3) / (24*n).

A335455 Number of compositions of n with some part appearing more than twice.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 11, 30, 69, 142, 334, 740, 1526, 3273, 6840, 14251, 29029, 59729, 122009, 248070, 500649, 1012570, 2040238, 4107008, 8257466, 16562283, 33229788, 66621205, 133478437, 267326999, 535146239, 1071183438, 2143604313, 4289194948, 8581463248

Offset: 0

Gus Wiseman, Jun 15 2020

nonn

Also the number of compositions of n matching the pattern (1,1,1).

A composition of n is a finite sequence of positive integers summing to n.

			The a(3) = 1 through a(6) = 11 compositions:
  (111)  (1111)  (1112)   (222)
                 (1121)   (1113)
                 (1211)   (1131)
                 (2111)   (1311)
                 (11111)  (3111)
                          (11112)
                          (11121)
                          (11211)
                          (12111)
                          (21111)
                          (111111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The case of partitions is A000726.
The avoiding version is A232432.
The (1,1)-matching version is A261982.
The version for patterns is A335508.
The version for prime indices is A335510.
These compositions are ranked by A335512.
Compositions are counted by A011782.
Combinatory separations are counted by A269134.
Normal patterns matched by compositions are counted by A335456.
Cf. A000740, A034691, A056986, A080599, A106356, A181796, A232464, A238279, A333755.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Max@@Length/@Split[Sort[#]]>=3&]],{n,0,10}]

a(n > 0) = 2^(n - 1) - A232432(n).

A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 3, 4, 3, 0, 1, 1, 2, 3, 4, 5, 4, 0, 1, 1, 2, 3, 5, 6, 7, 5, 0, 1, 1, 2, 3, 5, 6, 9, 9, 6, 0, 1, 1, 2, 3, 5, 7, 10, 12, 13, 8, 0, 1, 1, 2, 3, 5, 7, 10, 13, 16, 16, 10, 0, 1, 1, 2, 3, 5, 7, 11, 14, 19, 22, 22, 12, 0

Offset: 0

Ilya Gutkovskiy, May 11 2017

A(n,k) is the number of partitions of n in which no parts are multiples of k.

A(n,k) is also the number of partitions of n into at most k-1 copies of each part.

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  0,  1,  1,  1,  1,  1,  ...
  0,  1,  2,  2,  2,  2,  ...
  0,  2,  2,  3,  3,  3,  ...
  0,  2,  4,  4,  5,  5,  ...
  0,  3,  5,  6,  6,  7,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Partition Function b_k
Index entries for sequences related to partitions

Columns k=1-13 give: A000007, A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546, A341714.
Main diagonal gives A000041.
Mirror of A061198.
Cf. A061199, A210485.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k*i*(i+1)/2[0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
A:= (n, k)-> b(n$2, k-1)[1]:
seq(seq(A(n, 1+d-n), n=0..d), d=0..16);  # Alois P. Heinz, Oct 17 2018

Table[Function[k, SeriesCoefficient[Product[(1 - x^(i k))/(1 - x^i), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[x^k, x^k]/QPochhammer[x, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

A264905 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 6, 7, 8, 13, 16, 18, 26, 29, 38, 49, 58, 68, 90, 101, 125, 156, 181, 214, 263, 304, 358, 435, 505, 589, 701, 812, 939, 1115, 1275, 1485, 1736, 1991, 2286, 2667, 3038, 3489, 4028, 4588, 5240, 6036, 6833, 7787, 8904, 10078, 11429, 13020, 14698

Offset: 0

Vaclav Kotesovec, Nov 28 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000009, A000726, A001935, A100405, A266647, A266648, A266649, A266650, A266686, A275820.

nmax = 100; CoefficientList[Series[Product[1+x^k+x^(3*k), {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - Vaclav Kotesovec, Jan 05 2016

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Gus Wiseman, Mar 22 2025

nonn

Also numbers with a unique factorization into prime powers with distinct sums of prime indices.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The prime indices of 270 are {1,2,2,2,3}, and there are two multiset partitions into constant multisets with distinct sums: {{1},{2},{3},{2,2}} and {{1},{3},{2,2,2}}, so 270 is not in the sequence.
The prime indices of 300 are {1,1,2,3,3}, of which there are no multiset partitions into constant multisets with distinct sums, so 300 is not in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, of which there is only one multiset partition into constant multisets with distinct sums: {{1},{1,1},{3},{2,2}}, so 360 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}

For distinct blocks instead of block-sums we have A004709, counted by A000726.
Twice-partitions of this type are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
These are the positions of 1 in A381635.
For no choices we have A381636 (zeros of A381635), counted by A381717.
For strict instead of constant blocks we have A381870, counted by A382079.
Partitions of this type (unique into constant with distinct) are counted by A382301.
Normal multiset partitions of this type are counted by A382203.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower), A265947.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A001222, A003963, A005117, A047966, A050361, A293511, A300385, A381716.
Cf. A006171, A279784, A295935.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Length[Select[pfacs[#],UnsameQ@@hwt/@#&]]==1&]

A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 2, 3, 0, 2, 4, 4, 5, 0, 3, 5, 6, 6, 7, 0, 4, 7, 9, 10, 10, 11, 0, 5, 9, 12, 13, 14, 14, 15, 0, 6, 13, 16, 19, 20, 21, 21, 22, 0, 8, 16, 22, 25, 27, 28, 29, 29, 30, 0, 10, 22, 29, 34, 37, 39, 40, 41, 41, 42, 0, 12, 27, 38, 44, 49, 51, 53, 54, 55, 55, 56, 0, 15, 36

Offset: 0

Henry Bottomley, Apr 20 2001

			T(2,4) = 4 since the possible partitions of 4 are on the first definition (no term more than twice) 1+1+2, 2+2, 1+3, or 4 and on the second definition (no term a multiple of 3) 1+1+1+1, 1+1+2, 2+2, or 4.
Triangle T(n,k) begins:
1, 0, 0, 0, 0, 0,  0,  0,  0,  0, ...
   1, 1, 2, 2, 3,  4,  5,  6,  8, ...
      2, 2, 4, 5,  7,  9, 13, 16, ...
         3, 4, 6,  9, 12, 16, 22, ...
            5, 6, 10, 13, 19, 25, ...
               7, 10, 14, 20, 27, ...
                  11, 14, 21, 28, ...
                      15, 21, 29, ...
                          22, 29, ...
                              30, ...

Alois P. Heinz, Columns k = 0..140, flattened

Rows effectively include A000007, A000009, A000726, A001935, A035959.
Main diagonal is A000041.
A061198 is the same table but includes cases where n>k.
T(n,2*n) gives: A232623.

b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(k$2, n):
seq(seq(T(n, k), n=0..k), k=0..12);  # Alois P. Heinz, Nov 27 2013

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1, k], {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[k, k, n]; Table[Table[T[n, k], {n, 0, k}], {k, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

A266650 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)) / (1 - x^k).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 34, 53, 82, 123, 181, 263, 379, 537, 754, 1047, 1444, 1972, 2675, 3601, 4820, 6408, 8473, 11141, 14580, 18985, 24611, 31765, 40839, 52294, 66719, 84819, 107474, 135731, 170892, 214518, 268524, 335190, 417308, 518212, 641948, 793324, 978157

Offset: 0

Vaclav Kotesovec, Jan 02 2016

nonn

Convolution of A266686 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A100405, A266647, A266648, A266649, A266686.

nmax = 50; CoefficientList[Series[Product[(1+x^k-x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(6*c + Pi^2) * exp(sqrt((4*c + 2*Pi^2/3)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 + exp(-x) - exp(-3*x)) dx = 0.59698046904738615106237970379036510874974380079287087827737... . - Vaclav Kotesovec, Jan 05 2016

A035360 Number of partitions of n into parts 3k or 3k+1.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 7, 8, 11, 15, 17, 23, 30, 35, 44, 57, 66, 82, 103, 121, 146, 181, 211, 253, 308, 360, 425, 513, 596, 700, 834, 969, 1127, 1333, 1541, 1786, 2093, 2415, 2781, 3241, 3723, 4273, 4946, 5669, 6476, 7461, 8519, 9705, 11123, 12669, 14379, 16418

Offset: 0

Olivier Gérard

nonn

Euler transform of period 3 sequence [ 1, 0, 1, ...]. - Kevin T. Acres, Apr 28 2018

			1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 8*x^8 + 11*x^9 + ...

Robert Price, Table of n, a(n) for n = 0..1000

Cf. A032766, A132463, A000726, A035361.

nmax = 50; CoefficientList[Series[Product[1/((1 - x^(3*k))*(1 - x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 23 2015 *)
nmax = 52; kmax = nmax/3;
s = Flatten[{Range[0, kmax]*3}~Join~{Range[0, kmax]*3 + 1}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

a(n) ~ Gamma(1/3) * exp(2*Pi*sqrt(n)/3) / (4 * sqrt(3) * Pi^(2/3) * n^(11/12)). - Vaclav Kotesovec, Aug 23 2015

A035361 Number of partitions of n into parts 3k or 3k+2.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 2, 5, 5, 6, 9, 11, 11, 18, 19, 23, 31, 36, 40, 56, 60, 73, 92, 105, 121, 155, 170, 204, 247, 280, 325, 397, 440, 521, 615, 695, 805, 954, 1063, 1244, 1442, 1626, 1873, 2176, 2431, 2813, 3221, 3623, 4146, 4751, 5309, 6086, 6905, 7746, 8807

Offset: 0

Olivier Gérard

nonn

Euler transform of period 3 sequence [ 0, 1, 1, ...]. - Kevin T. Acres, Apr 28 2018

			1 + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 5*x^9 + 6*x^10 + ...

Robert Price, Table of n, a(n) for n = 0..1000

Cf. A007494, A132462, A000726, A035360.

nmax = 50; CoefficientList[Series[Product[1/((1 - x^(3*k))*(1 - x^(3*k-1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 23 2015 *)
nmax = 55; kmax = nmax/3;
s = Flatten[{Range[0, kmax]*3}~Join~{Range[0, kmax]*3 + 2}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

a(n) ~ Gamma(2/3) * exp(2*Pi*sqrt(n)/3) / (4 * sqrt(3) * n^(13/12) * Pi^(1/3)). - Vaclav Kotesovec, Aug 23 2015

cp's OEIS Frontend

A257989 The crank of the partition having Heinz number n.

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A266648 Expansion of Product_{k>=1} (1 + x^(3*k)) / (1 - x^k).

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A335455 Number of compositions of n with some part appearing more than twice.

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A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

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Maple

Mathematica

Formula

A264905 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).

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Mathematica

Formula

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

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Mathematica

A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).

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Maple

Mathematica

A266650 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)) / (1 - x^k).

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